To find the exact value of cos(5Ï€/8) using the half angle identity, we start by rewriting 5Ï€/8 as the product of 1/2 and 5Ï€/8:
5Ï€/8 = (1/2)(5Ï€/8)
Now, we can use the half angle identity for cosine, which states that:
cos(θ/2) = ±√[(1 + cos(θ))/2]
Let's apply this identity to our expression:
cos(5Ï€/8) = cos((1/2)(5Ï€/8))
Now, using the half angle identity:
cos((1/2)(5π/8)) = ±√[(1 + cos(5π/8))/2]
To determine the sign (±), we need to consider the quadrant where 5π/8 lies.
In the first quadrant, the value of cos(θ) is positive.
In the second quadrant, the value of cos(θ) is negative.
Since 5π/8 is between π/2 and π, it lies in the second quadrant, and therefore cos(5π/8) is negative.
Now, let's plug the value of cos(5Ï€/8) into the half angle identity equation:
cos((1/2)(5π/8)) = -√[(1 + cos(5π/8))/2]
We still need to find the value of cos(5Ï€/8), which we can obtain using the half angle identity again.
Let's define a new angle, let's say α, which satisfies the equation:
(1/2)(5π/8) = α/2
Simplifying the equation, we get:
5π/8 = α
Now, we can rewrite cos((1/2)(5π/8)) in terms of α:
cos((1/2)(5π/8)) = cos(α/2)
Applying the half angle identity to α/2, we have:
cos(α/2) = ±√[(1 + cos(α))/2]
Since α = 5π/8, we can substitute it back:
cos(5π/8) = ±√[(1 + cos(5π/8))/2]
We now have an equation involving cos(5Ï€/8) and we can solve it.
To do so, let's multiply both sides by 2:
2cos(5π/8) = ±√(1 + cos(5π/8))
Next, square both sides to eliminate the radical:
(2cos(5π/8))^2 = (±√(1 + cos(5π/8)))^2
4cos^2(5Ï€/8) = 1 + cos(5Ï€/8)
Now, let's rearrange the equation:
4cos^2(5Ï€/8) - cos(5Ï€/8) - 1 = 0
This is a quadratic equation in terms of cos(5Ï€/8). We can solve it by factoring or using the quadratic formula:
cos(5π/8) = [-(-1) ± √((-1)^2 -4(4)(-1))]/(2(4))
cos(5π/8) = (1 ± √(1 + 16))/8
cos(5π/8) = (1 ± √17)/8
Therefore, the exact value of cos(5Ï€/8) in terms of radicals is:
cos(5π/8) = (1 ± √17)/8